About the Dirac Operator

Adriana Turtoi

Dedicated to the memory of Radu Rosca (1908-2005)

Abstract. We apply a result of Kim about the eigenvalue estimation of the Dirac operator on a Riemannian compact spin manifold (M, g), considering M = N ×S1, where N is a Riemannian compact spin manifold admitting a parallel vector ﬁeld. We show that the lower bounds given in a theorem of Hijazi and Zhang for the eigenvalues of the so called (submanifold) twisted Dirac operator DH in the case when H 6= 0 is true for H = 0 also. As an example, we consider every spin K¨ahlermanifold as a totally geodesic submanifold of its twistor space and we study its twisted Killing spinors.

Mathematics Subject Classiﬁcation: 53C27, 58B40. Key words: Dirac operator, eigenvalues estimation.

1 Introduction

In general, the term ”Dirac operator” is used to refer to any first-order operator which factorizes the ”Laplacian” for a given quadratic space of arbitrary signature. In 1928, Dirac [10] introduced a first-order linear operator in order to express the ∂2 ∂2 ∂2 ∂2 square root of the wave operator ut = 2 − ( 2 + 2 + 2 ). ∂x0 ∂x1 ∂x2 ∂x3 Dirac constructed this first-order operator using the Dirac algebra which is a particular realization of the Clifford algebra associated to the real quadratic form of signature (1,3). We briefly review the construction of the Dirac algebra, using the same method that Dirac did. Let µ ¶ µ ¶ µ ¶ µ ¶ 1 0 1 0 0 −i 0 1 σ = , σ = , σ = , σ = 0 0 1 1 0 −1 2 i 0 3 1 0 denote the Pauli matrices. The Pauli matrices are used to construct the Dirac γ−matrices µ ¶ µ ¶ 0 σ0 0 σj γ0 = , γj = , j = 1, 2, 3. σ0 0 −σj 0

2 2 2 2 The γ−matrices satisfy γ0 = I, γ1 = γ2 = γ3 = −I, γjγk = −γkγj, j 6= k. Balkan∗ Journal of Geometry and Its Applications, Vol.11, No.1, 2006, pp. 121-130. °c Balkan Society of Geometers, Geometry Balkan Press 2006. 122 Adriana Turtoi

As a differential operator on C4−valued (i.e. spinor-valued) function on an open 4 ∂ ∂ ∂ ∂ set U ⊆ R , the Dirac operator D = γ0 + γ1 + γ2 + γ3 has the desired ∂x0 ∂x1 ∂x2 ∂x3 property D2 = ut. Not long after Dirac’s original construction, Brauer and Weyl generalized it to any arbitrary finite-dimensional quadratic space of arbitrary signature, using the universal Clifford algebra associated with a real non-degenerate quadratic space [9]. The natural operations for vector spaces with quadratic forms carry over to vector bundles with metrics. In particular, suppose that π : TM → M is the tangent vector −1 bundle of the Riemannian manifold (M, g). Then in each fibre Tx(M) = π (x), the quadratic form gx can be used to construct the associated Clifford algebra Cl(gx). The result is the a bundle Cl(M) → M of algebras over M called the Clifford bundle of M. In the light of representation theory of Clifford algebra, it is natural to ask whether one can also find a vector bundle S(M) → M with the property that each fibre Sx is an irreducible module over Cl(gx). The answer is in general no. But, with some obstructions, if M is an oriented Riemannian manifold and the second Stiefel -Whitney class w2(M) is vanish, the vector bundle S(M) → M exist. Then, this will lead to the notion of a spin structure and spin manifold M, using also the theory of morphism fibre bundles. Let (M, g) be a compact Riemannian spin manifold and S(M) a spinor fibre bundle on M. We can define a canonical first-order differential operator D : Γ(S(M)) → Pn Γ(S(M)) called the Dirac operator on M, by setting Dσ = i=1 ei.∇ei σ, at x ∈ M, where {e1, ..., en} is an orthonormal basis of TxM, where ∇ denotes the covariant derivative on S(M) determined by the connection and where ”.” denotes the Clifford module multiplication. The operator D2 is called the Dirac laplacian. In 1963, AndréLichnerowicz was the first to consider estimating eigenvalues of Dirac operator. He was also the first to provide a rough approximation of this eigenvalues. By integration, from the Schrödinger-Lichnerowicz formula [21] s (1.1) D2 = ∇∗∇ + 4 it is clear that all eigenvalues λ of D must satisfy the inequality s (1.2) λ2 ≥ 0 4 where s0 = min{s(x) | x ∈ M} is the minimum of the scalar curvature s. Of course, this is interesting only if the scalar curvature is positive but then the estimate (1.2) is never sharp. In 1980, the first sharp estimation was given by Friedrich [12] who proved [11] that n (1.3) λ2 ≥ s . 4(n − 1) 0 Friedrich’s proof is based on a modification of the Levi-Civita connection and a formula similar to (1.1). The boundary of this estimate is characterized by the existence of a non-trivial real Killing spinor on M. The manifolds with non-trivial real Killing spinors have constant positive scalar curvature. They are furthermore Einstein and do not admit any parallel k− form for k 6= 0, n. This shows that the estimation (1.3) cannot be sharp in some situations. About the Dirac operator 123

2 n In 1986, Hijazi [17] proved that in dimension n ≥ 3, λ ≥ 4(n−1) µ1 where µ1 is n−1 the first eigenvalue of the Yamabe operator Y = 4 n−2 ∆ + s. Ch. Bärshowed [3], [4] that for a closed and connected Riemannian spin manifold 2 2πχ(M) of dimension n = 2, we have λ ≥ area(M) , where χ(M) is the Euler Poincarénumber of M. B. Alexandrov, G. Grantcharov and S. Ivanov [2] showed that if M admits a parallel 1− form, then

n − 1 s (1.4) λ2 ≥ 0 . n − 2 4

Moroianu and Ornea [24] proved that the same holds true under the weaker as- sumption that the 1− form is harmonic and has constant length. k i i In [1] B. Alexandrov proved that if TM = ⊕i=1T , where T are parallel distributions of dimension ni, with n1 < ... < nk, then n s (1.5) λ2 ≥ k 0 nk − 1 4 and this result is a generalization of one of Kim [19] for k = 2. E. Witten [30] has introduced the hypersurface Dirac operator to prove the positive mass theorem. The spinorial background that has developed to extend the classical estimates to hypersurfaces has now become a powerful tool to investigate extrinsec geometry and manifolds with boundary problems. Using Hijazi-Zhang’s theorem, in [15], [14] lower bounds are given for the eigenvalues of the so called (submanifold) twisted Dirac operator DH and are discussed their limiting cases if the mean curvature H 6= 0. We showed that this study is very natural, because when the considered spin submanifold is minimal, and therefore H = 0, Hijazi-Zhang’s theorem is true as well. Furthermore, the corresponding (real) twisted Killing spinors play the same role as the Killing spinors for spin manifolds. As an example, we have considered every spin K¨ahlermanifold as a totally geodesic submanifold of its twistor space and we study its twisted Killing spinors.

2 First application

Let (N, g) be a n− dimensional Riemannian manifold. In [29] we have showed:

1 1 Theorem 2.1. Let ξ be a global ﬁeld on N. Let Di : N × S → Tp(N × S ), i = 1, 2 1 2 be the maps D1 : p = (x, a) → Tp ,D2 : p = (x, a) → Tp where

1 Tp = Sp{(−ξp, 1), (ξp, 1)} 2 Tp = {(X, 0) | X ∈ Tp(N), gp(ξp,X) = 0}

Tp(N) being the tangent space of N on the point p. Then, the maps D1,D2 gives rise respectively to two smooth distributions T 1 and T 2 on NXS1 of dimension 2 and n−1 respectively. 124 Adriana Turtoi

Proposition 2.1. The distributions given by D1,D2 are parallel if the vector field ξ is parallel. Let (N, g) be a Riemannian, compact n−dimensional spin manifold, n ≥ 2, with fixed spin structure and ξ a parallel vector field on N. On the compact Riemannian 1 1 spin manifold (N × S ,G), where G = g + g0, g0 being the standard metric of S , there is the global parallel vector field (ξ, 1). This fact allows as to use the formula (1.4), and we have the following estimation for an eigenvalue λ of the Dirac operator on N × S1 n s + 1 (2.1) λ2 ≥ 0 , n − 1 4 where s0 is the minimum of the scalar curvature s of N. In the other hand, the tangent space T (N × S1) = T 1 ⊕ T 2, where T 1,T 2 are parallel distributions of dimension 2, n − 1, respectively. Then the first eigenvalue λ of the Dirac operator satisfies, according with (1.5), the inequality ½ n−1 s0+1 2 n−2 4 if n ≥ 3 (2.2) λ ≥ s0+1 2 if n = 2. This last estimation is better than (2.1). The only compact, orientable and therefore spin surface having a global parallel vector field is the two dimensional torus T and we may apply the formula (2.2) for it.

3 Second application

Let (M,Gf ) be a Riemannian m + n− dimensional spin manifold and let M be an immersed oriented m− dimensional submanifold in Mf with the induced Riemannian structure g = G|M . Assume that (M, g) is spin. Denote by NM the normal vector bundle of M in M.f There exists also a spin structure on NM [23]. Let ΣM, ΣN and ΣMf be the spinor bundles over M,NM and Mf respectively. f The restricted spinor bundle S := ΣM|M may be considered and is possible to identify it with Σ =: ΣM ⊗ ΣN [14] if mn is even. If m and n are both odd, one has to take Σ as the direct sum of two copies of that bundle. Denote by (e1, ..., em, υ1, ..., νn) a positively oriented local orthonormal basis of f T M|M such that (e1, ..., em) (resp. (υ1, ..., νn)) is a positively oriented local orthonormal basis of TM (respectively of NM). If ∇e denotes the Levi-Civita connection of (M,Gf ), then for all X ∈ Γ(TM), for all A ∈ Γ(TN) and i = 1, ..., m, the Gauss ∗ formula may be written, as ∇e i(X + A) = ∇i(X + A) + h(ei,X) − h (ei,A) where M N ∗ ∇i(X + A) = ∇ei X + ∇ei A, h (ei,.) is the transpose of second fundamental form h e e viewed as a linear map from TM to NM, and ∇i stands for ∇ei . Denote also by ∇e and ∇ the induced spinorial covariant derivative on Γ(S). There- fore, on Γ(S) we have:   (∇ΣM ⊗ Id + Id ⊗ ∇ΣN ) ⊕ (∇ΣM ⊗ Id + Id ⊗ ∇ΣN ) if m, n odd (3.1) ∇e =  ∇ΣM ⊗ Id + Id ⊗ ∇ΣN otherwise About the Dirac operator 125

The spinorial Gauss formula [7] is

1 X2m ∇e ϕ = ∇ ϕ + e .h .ϕ, (∀)ϕ ∈ Γ(S). i i 2 j ij j=1

The following submanifold Dirac’s operators [14], may be introduced:

Xm Xm 1 (3.2) De = e · ∇e ,D = e .∇ ,D = (−1)nω · D + H.ω , i i i i H ⊥ 2 ⊥ i=1 i=1

Pm where we have denoted by H =: i=1 h(ei, ei) the mean curvature vector ﬁeld and where: ½ ωn for n even, ω⊥ = iωn for n odd,

ωn denoting the complex volume form:

[ n+1 ] ωn = i 2 ν1...νn.

2 n In both cases ( ω⊥) = (−1) . We consider that (M, g) is a minimal submanifold of (M,Gf ), therefore we have H = 0 and for all ϕ ∈ Γ(S)

n (3.3) DH ϕ = (−1) ω⊥Dϕ,

Recall that there exists a hermitian inner product on Γ(S), denoted by < ., . >, f such that Clifford multiplication by a vector of T M|M is skew-symmetric. In the following, we write (., .) = Re(< ., . >). N For any spinor field ϕ ∈ Γ(S) is defined the application Rϕ : Mϕ → R with

Xm N N ϕ R := 2 (ei.ej.Id ⊗ R ϕ, )(3.4) ϕ eiej | ϕ |2 i,j=1

N and Mϕ := {x ∈ M | ϕ(x) 6= 0}, where Reiej stand for spinorial normal curvature tensor [14].

Theorem 3.1. Let (M, g) be a compact m − dimensional immersed, minimal spin submanifold in the spin manifold (M,Gf ). Then, denoting the scalar curvature of (M, g) by R, we have

2 1 N (3.5) λ ≥ inf (R0 + Rϕ ), Mϕ 4 where R0 = infM R and λ is an eigenvalue of the twisted Dirac operator DH . 126 Adriana Turtoi

Proof. Using (3.1), the Schr¨odinger-Lichnerowicz formula for the twisted Dirac operator DH becomes

Xm Xm 2 R 1 N (3.6) D ϕ = (Id ⊗ Id)ϕ + eiejId ⊗ R ϕ − ∇i∇iϕ H 4 2 eiej i,j=1 i=1 and, therefore, we obtain

R + RN (3.7) (D2 ϕ, ϕ) = ϕ | ϕ |2 + | ∇ϕ |2 . H 4 Suppose that ϕ ∈ Γ(S) is a non-zero eigenvalue spinor of the twisted Dirac operator, so that DH ϕ = λϕ. Therefore, (3.7) gives

R + RN (λ2 − ϕ ) | ϕ |2=| ∇ϕ |2 . 4 Assuming by absurd that

R + RN λ2 − ϕ < 0 4 then it results ϕ = 0 in contradiction with the hypothesis. Hence the inequality is veriﬁed. The estimation (3.5) is not optimal. We have the following:

Theorem 3.2. Let (M, g) be a compact m − dimensional immersed minimal spin submanifold in the spin manifold (M,Gf ) and λ an eigenvalue of the twisted Dirac operator DH . Then, if

N R + Rϕ > 0 i) the following inequality holds

m R + RN (3.8) λ2 ≥ inf ϕ . m − 1 Mϕ 4 q 1 m N ii) If λ = ± 2 m−1 infMϕ (R + Rϕ ) is an eigenvalue corresponding to the eigenspinor ϕ for the twisted Dirac operator DH , then the following equations are satisﬁed: r 1 m ∇ ϕ + (R + RN )X.ω .ϕ = 0(3.9) X 2m m − 1 ϕ ⊥

r 1 m ∇ ϕ − (R + RN )X.ω .ϕ = 0(3.10) X 2m m − 1 ϕ ⊥ for all X ∈ Γ(TM). About the Dirac operator 127

Proof. i) Let ϕ be an eigenspinor for the twisted Dirac operator DH , so that

DH ϕ = λϕ. We consider the modiﬁed connection

λ [ n ] λ (3.11) ∇ m = ∇ + (−1) 2 e .ω , (∀)i = 1, ..., m. i i m i ⊥ We can easily compute

m X λ λ 2 λ 2 2 λ 2 (3.12) | ∇ m ϕ | = (∇ m ϕ, ∇ m ϕ) =| ∇ϕ | − | ϕ | . i i m i=1 The equality (3.12) is a consequence of the fact that the Cliﬀord multiplication with ei, i = 1, ..., m, and γj, j = 1, ..., m is orthogonal and so Xm [ n ]+1 [ n ]+1 2 ( ∇iϕ, eiω⊥ϕ) = (−1) 2 (DH ϕ, ϕ) = (−1) 2 λ | ϕ | . i=1 On the other hand, using Schr¨odinger-Lichnerowicz formula (3.6), (3.12) and (3.7) we obtain N λ 2 R + Rϕ m − 1 2 2 λ 2 ((D − ) ϕ, ϕ) = [ − λ ] | ϕ | + | ∇ m ϕ | H m 4 m2 and by a direct calculus λ (m − 1)2 ((D − )2ϕ, ϕ) = λ2 | ϕ |2 . H m m2 Comparing the last two relations, we obtain

N R + Rϕ m − 1 2 2 λ 2 (3.13) [ − λ ] | ϕ | + | ∇ m ϕ | = 0. 4 m

Because ϕ 6= 0, thisq equality (3.13) implies (3.8). 1 m N N ii) If λ = ± 2 m−1 infMϕ (R + Rϕ ), the equality (3.13) implies that R + Rϕ is λ 2 λ constant, Mϕ = M and | ∇ m ϕ | = 0. So, ∇ m ϕ = 0, and the definition (3.11) implies respectively the equations (3.9), (3.10). Remark 3.1. Compare theorem 4 with Hijazi-Zhang’s theorem [15], [14], which is proved when H 6= 0, note that this is also true for H = 0, as shown in formula (3.8). Let (M, g) be an oriented Riemannian manifold of dimension m = 2p and Z(M) its corresponding twistor space. It is well know that the twistor space Z(M) may be endowed with a natural metric G such that π : Z(M) → M results to be a Riemannian submersion and the vertical and horizontal parts are orthogonal. Assume that (M, g) is a Kählermanifold. This hypotheses implies that U(n) is the structure group of its principal fibre bundle of orthogonal frames and, moreover, the connection form of the Levi-Civita connection of the metric g is u(n)−valued. We have showed [28] 128 Adriana Turtoi

Theorem 3.3. Let (M, g) be a 2p− dimensional K¨ahlermanifold and Z(M) its corresponding space of twistors. Then, there exists (M,Gc ) an immersed oriented submanifold in the twistor manifold (Z(M),G), such that π : Mc → M is an isometric transformation.

Remark 3.2. Via the projection π we may identify the manifolds (M,Gc ) and (M, g).

We may consider the normal vector fibre bundle NM of M in Z(M), with respect to the metric G. Assume that (M, g) is spin. It is proved [25], [26] that if M admits a spin structure, then its twistor space Z(M) also admits a spin structure. There exists also a spin structure on NM [23]. Let ΣM, ΣN and ΣZ(M) be the spinor bundles over M,NM and Z(M) respectively. The restricted spinor bundle S := ΣZ(M)|M may be considered and is possible to be identified with Σ =: ΣM ⊗ ΣN [14]. We have showed [28] that if M is a Kählermanifold, then Mc is a totally geodesic submanifold of Z(M). Therefore, the theorems 3 and 4 holds with m = 2p and n = p(p − 1), because H = 0. Recall [14] that a non-zero section ϕ ∈ Γ(S) satisfying µ (3.14) ∇ ϕ = − X.ω .ϕ, (∀)X ∈ Γ(TM), X m ⊥ for a given real constant µ is called a twisted (real) Killing spinor. If ϕ ∈ Γ(S) is a twisted Killing spinor, so that we have (3.14), then it is easy to check that DH ϕ = µϕ and so, ϕ is an eigenspinor of the twisted Dirac operator DH , corresponding to the eigenvalue µ. We have:

Theorem 3.4. Let (M, g) be a K¨ahlerspin manifold of dimension m = 2p, with a twisted Killing spinor ϕ so that we have (3.14). Then the manifold (M, g) is an Einstein space. Moreover the scalar curvature function R is positive and

1 m (3.15) µ2 = R. 4 m − 1 Remark 3.3. An analysis similar to the one above can be carried out for a Rie- mannian spin manifold considered as a submanifold of its manifold of euclidean inner products [27], but this submanifold is not minimal.

References

[1] B. Alexandrov, The ﬁrst eigenvalue of the Dirac operator on locally reducible Riemannian manifolds, arXiv:math. DG/0502536.

[2] B. Alexandrov, G. Grantcharov, S. Ivanov, An estimate for the ﬁrst eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one-form, J. Geom. Phys. 28 (1998), 263-270. About the Dirac operator 129

[3] C. B¨ar, Das Spectrum von Dirac-Operatoren, Doktorarbeit, Universit¨atBonn, 1991, Bonner Math. Sch., 217.

[4] C. B¨ar, Lower eigenvalue estimates for Dirac operators, Math. Ann., 293 (1992), 39-46.

[5] B¨arCh., Real Killing Spinors and Holonomy, Com. Math. Phys. 154, (1993), 525-576.

[6] Baum Helga, Friedrich Th., Grunewald R.,Kath Ines, Twistors and Killing Spinors on Riemannian Manifolds, Teubner Texte zur Mathematik, 124, B. G. Teubner Verlagsgesellschaft, Stuttgart, (1991).

[7] B¨arC., Extrinsec Bound for Eigenvalues of the Dirac Operator, Ann. Glob. Anal. Geom. 16, (1988), 573-596.

[8] Ch. B¨ar,M. Dahl, The ﬁrst Dirac eigenvalue on manifolds with positive scalar curvature, arXiv: math. DG/0305277 v1, 19 May 2003.

[9] R. Brauer, H. Weyl, Spinors in n Dimensions, Amer. J. of Math. 57 (1935), 425-449.

[10] P.A.M. Dirac, The quantum theory of the electron, I,II, Proc Roy. Soc. London, A117 (1928), 610-624; A118 (1928), 351-361.

[11] T. Friedrich, Der erste Eigenwert des Dirac-Operator einer kompakten Rieman- nishen Mannigfaltigkeit nicht-negativer Kr¨ummung, Math. Nachr. 97 (1980), 117-146.

[12] Th. Friedrich, Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol. 25, American Mathematical Society, Providence, Rhode Island, 2000.

[13] T. Friedrich, Der erste Eigenwert des Dirac Operators einer kompakten, Rie- mannischen Mannigfaltigkeit nichtnegativer Skalarkr¨ummung, Math. Nachr. 97 (1980), 117-146. Mathematical Society, Providence, Rhode Island, (2000).

[14] Ginoux N., Morel B., On eigenvalue estimates for the submanifold Dirac operator, Internat. J. Math., 13, 5, (2002), 533-548.

[15] Hijazi O.,Zhang X., Lower bounds for the Eigenvalues of Dirac Operator, Part II. The Submanifold Dirac Operator, Ann. Glob. Anal. Geom. 20, 2, ( 2001), 163-181.

[16] O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151-162.

[17] O. Hijazi, Opérateurs de Dirac sur les variétésRiemanniennes. Minoration des valeurs propres, Thèsede 3- èmecycle, Ecole Polytechnique, 1984. 130 Adriana Turtoi

[18] K. D. Kirchberg, An estimation for the ﬁrst eigenvalue of the Dirac operator on closed K¨ahlermanifolds with positive scalar curvature, Ann. Glob. Anal. Geom. 4 (1986), 291-326.

[19] E. C. Kim, Lower bounds of the Dirac eigenvalues on Riemannian product manifolds, arXiv: math. DG/0402427.

[20] S. Kobayashy, K. Nomizu, Foundations of Diﬀerential Geometry, I, II, Inter- science Publishers, New York, London, 1963, 1969.

[21] A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris 257 (1963), 7-9.

[22] H. B. Lawson Jr., M.L. Michelsohn, Spin Geometry, Princeton University Press, Princeton, New Jersey, 1989. [23] J. Milnor, Remarks concerning spin manifolds, Princeton Univ. Press, (1965), 55-62. [24] A. Moroianu, l. Ornea, Eigenvalue estimates for the Dirac operator and harmonic 1-forms of constant length, C. R. Math. Acad. Sci. Paris 338 (2004), 551-564. [25] A. Nannicini, Vanishing theorems for twisor spaces, Ann. Sc. Norm. Sup. di Pisa,19, 4,(1992), 183 -205. [26] A.Turtoi, Prolongation of Spin Structures on Twistor Space, An. Univ. Bucure¸sti, 47, 2, (1998), 235-246. [27] A. Turtoi, The manifold of euclidean inner products, Analele Universit˘at¸ii Bu- cure¸sti,52, 1, (2003), p. 107–120. [28] A. Turtoi, Twisted Dirac operators on minimal submanifolds ( send for publica- tion) [29] A. Turtoi, First eigenvalues of the Dirac operator, Proceedings of 9th National Conference of the Romanian Mathematical Society, Lugoj, May,2005 (in print)) [30] E. Witten, A new proof of the positive energy theorem, Commun. Math. Phys. 80 (1981), 381-402.

Author’s address: Adriana Turtoi University of Bucharest, Faculty of Mathematics and Informatics, Str. Academiei 14, RO-70109, Bucharest, Romania email: [email protected]